1 Department of Mathematics, Science and Technology, Aarhus University2 Chalmers University of Technology and Mathematical Sciences, Göteborg3 Department of Mathematics, Science and Technology, Aarhus University
We explicitly construct a finite number of discrete components in the restriction of complementary series representations of rank one semisimple groups $G$ to rank one subgroups $G_1$. For this we use the realizations of complementary series representations of $G$ and $G_1$ on Sobolev spaces on the nilpotent radicals $N$ and $N_1$ of the minimal parabolics in $G$ and $G_1$, respectively. The groups $N$ and $N_1$ are of H-type and we construct explicitly invariant differential operators between $N$ and $N_1$. These operators induce the projections onto the discrete components. Our construction of the invariant differential operators is carried out uniformly in the framework of H-type groups and also works for those H-type groups which do not occur as nilpotent radical of a parabolic subgroup in a semisimple group.
Journal of Geometric Analysis, 2016, Vol 26, Issue 1, p. 118-142